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First of all, there is a lot of misunderstanding about the graph search and tree search. The difference between these two is not about graphs and trees. They have two different algorithms. You can find the difference in this answer to the question What is the difference between graph-search and tree search?.
Now, I want to be sure that I have understood these two searches well. We have a graph search and a tree search for each searching algorithm. For example, BFS graph search and BFS tree search. Or DFS graph search and DFS tree search (DFS tree search is not complete, for example). Iterative deepening graph search and iterative deepening tree search. Is this correct?